Mastering the Sine and Cosine Rules for Your Mathematics Examination

What the examiner is testing

Examiners are testing your ability to select and correctly apply the sine and cosine rules to find unknown sides or angles in non-right-angled triangles. Marks are awarded for choosing the correct formula, substituting values accurately, and performing the algebraic rearrangement to isolate the unknown.

The method

1. Identify the type of triangle: Confirm it is not a right-angled triangle. If it is, use SOHCAHTOA instead.
2. Label the triangle: Label the vertices with capital letters (e.g., A, B, C) and the sides opposite them with corresponding lowercase letters (a, b, c).
3. Determine which rule to use:
   • Sine Rule: Use if you have a "pair" (an angle and its opposite side) and one other piece of information (either another angle or another side).
     • To find a side: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
     • To find an angle: \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)
   • Cosine Rule: Use if you do not have a "pair". This typically means you have:
     • Two sides and the included angle (SAS) to find the third side.
     • All three sides (SSS) to find an angle.
     • To find a side: \( a^2 = b^2 + c^2 - 2bc \cos A \)
     • To find an angle: \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
4. Substitute known values: Carefully place the given numbers into the chosen formula.
5. Rearrange and solve: Isolate the unknown variable using algebraic manipulation.
6. Calculate the result: Use your calculator to find the numerical answer, ensuring it's to an appropriate degree of accuracy (e.g., one decimal place for angles, three significant figures for lengths, unless specified).

Worked example

Find the length of side \( x \) in the triangle below, correct to 3 significant figures.

$$ \begin{array}{c} \text{A triangle with angles } 40^\circ \text{ and } 65^\circ \text{ and a side of length } 10 \text{ cm opposite the } 65^\circ \text{ angle. The side opposite the } 40^\circ \text{ angle is labelled } x. \end{array} $$

1. Identify the type of triangle: It's not a right-angled triangle.
2. Label the triangle:
   Let \( A = 40^\circ \), \( a = x \)
   Let \( B = 65^\circ \), \( b = 10 \text{ cm} \)
   The third angle is \( C = 180^\circ - 40^\circ - 65^\circ = 75^\circ \).
3. Determine which rule to use: We have a pair (\( B = 65^\circ \) and \( b = 10 \text{ cm} \)) and we want to find side \( x \) (which is \( a \)) opposite angle \( A \). The Sine Rule is appropriate.
   To find a side: \( \frac{a}{\sin A} = \frac{b}{\sin B} \)
4. Substitute known values:
   \( \frac{x}{\sin 40^\circ} = \frac{10}{\sin 65^\circ} \)
5. Rearrange and solve:
   \( x = \frac{10 \times \sin 40^\circ}{\sin 65^\circ} \)
6. Calculate the result:
   \( x = \frac{10 \times 0.64278...}{0.90630...} \)
   \( x = \frac{6.4278...}{0.90630...} \)
   \( x = 7.0924... \text{ cm} \)
   \( x = 7.09 \text{ cm} \) (to 3 significant figures)

Sanity check: The angle \( 40^\circ \) is smaller than \( 65^\circ \), so the side opposite \( 40^\circ \) (which is \( x \)) should be smaller than the side opposite \( 65^\circ \) (which is \( 10 \text{ cm} \)). \( 7.09 \text{ cm} < 10 \text{ cm} \), so the answer is reasonable.

Worked example: a harder one

A triangle has sides \( AB = 8 \text{ cm} \), \( BC = 11 \text{ cm} \) and \( AC = 15 \text{ cm} \). Find the size of angle \( ABC \), correct to one decimal place.

Why the obvious first move fails: Some students might try to use the Sine Rule first. To use the Sine Rule, you need a pair (an angle and its opposite side). In this problem, we have three sides but no angles. Therefore, the Sine Rule cannot be used directly. We must use the Cosine Rule.

1. Identify the type of triangle: It's not a right-angled triangle.
2. Label the triangle:
   Let \( a = AC = 15 \text{ cm} \) (side opposite angle B)
   Let \( b = AB = 8 \text{ cm} \) (side opposite angle C)
   Let \( c = BC = 11 \text{ cm} \) (side opposite angle A)
   We need to find angle \( ABC \), which is angle \( B \).
3. Determine which rule to use: We have all three sides (SSS) and need to find an angle. The Cosine Rule is appropriate.
   To find an angle: \( \cos B = \frac{a^2 + c^2 - b^2}{2ac} \)
   Note: Be careful with the labelling for the Cosine Rule. If finding angle B, the side opposite it, b, is subtracted.
4. Substitute known values:
   \( \cos B = \frac{15^2 + 11^2 - 8^2}{2 \times 15 \times 11} \)
5. Rearrange and solve:
   \( \cos B = \frac{225 + 121 - 64}{330} \)
   \( \cos B = \frac{282}{330} \)
   \( \cos B = \frac{47}{55} \)
   \( B = \cos^{-1}\left(\frac{47}{55}\right) \)
6. Calculate the result:
   \( B = 31.002...^\circ \)
   \( B = 31.0^\circ \) (to one decimal place)

Sanity check: The smallest side is 8 cm, which is opposite angle B. The angle \( 31.0^\circ \) is an acute angle, and it's reasonable for it to be the smallest angle given it's opposite the smallest side.

Practice

1. A triangle has sides \( 7 \text{ cm} \) and \( 9 \text{ cm} \), and the angle between them is \( 70^\circ \). Find the length of the third side, correct to 3 significant figures.
2. In triangle PQR, \( PQ = 12 \text{ cm} \), \( QR = 15 \text{ cm} \) and angle \( PQR = 105^\circ \). Find the length of PR, correct to 3 significant figures.
3. In triangle XYZ, \( XY = 10 \text{ cm} \), angle \( YXZ = 48^\circ \) and angle \( XZY = 75^\circ \). Find the length of YZ, correct to 3 significant figures.
4. The diagram shows a quadrilateral ABCD. \( AB = 7 \text{ cm} \), \( BC = 9 \text{ cm} \), \( CD = 12 \text{ cm} \), \( AD = 10 \text{ cm} \). Angle \( ABC = 100^\circ \). Calculate the size of angle \( ADC \), correct to one decimal place.

Answers:
1. \( 9.51 \text{ cm} \)
2. \( 20.4 \text{ cm} \)
3. \( 8.01 \text{ cm} \)
4. Working for Q4:
* First, use the Cosine Rule in triangle ABC to find AC.
\( AC^2 = AB^2 + BC^2 - 2(AB)(BC) \cos(\angle ABC) \)
\( AC^2 = 7^2 + 9^2 - 2(7)(9) \cos(100^\circ) \)
\( AC^2 = 49 + 81 - 126 \cos(100^\circ) \)
\( AC^2 = 130 - 126(-0.1736...) \)
\( AC^2 = 130 + 21.88... \)
\( AC^2 = 151.88... \)
\( AC = \sqrt{151.88...} = 12.324... \text{ cm} \)
* Now, use the Cosine Rule in triangle ADC to find angle ADC.
Let \( \angle ADC = D \). We have sides \( AD = 10 \text{ cm} \), \( CD = 12 \text{ cm} \) and \( AC = 12.324... \text{ cm} \).
\( \cos D = \frac{AD^2 + CD^2 - AC^2}{2(AD)(CD)} \)
\( \cos D = \frac{10^2 + 12^2 - (12.324...)^2}{2 \times 10 \times 12} \)
\( \cos D = \frac{100 + 144 - 151.88...}{240} \)
\( \cos D = \frac{92.11...}{240} \)
\( \cos D = 0.3838... \)
\( D = \cos^{-1}(0.3838...) \)
\( D = 67.42...^\circ \)
Angle \( ADC = 67.4^\circ \) (to one decimal place).

The three mistakes that lose marks

1. Using the wrong rule: Attempting to use the Sine Rule when only three sides (SSS) or two sides and the included angle (SAS) are known.
   • Wrong answer produced: An inability to set up the equation, or an incorrect answer if a formula is misremembered. For example, trying to find an angle with SSS using sine rule leads to \( \frac{\sin A}{a} = \frac{\sin B}{b} \) with no angles given.
2. Incorrectly identifying the included angle or opposite side for the Cosine Rule: For \( a^2 = b^2 + c^2 - 2bc \cos A \), angle \( A \) must be the angle between sides \( b \) and \( c \), and side \( a \) must be opposite angle \( A \). Similarly for finding an angle.
   • Wrong answer produced: If you use \( \cos A = \frac{a^2 + c^2 - b^2}{2ac} \) when you should have used \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \), you will subtract the wrong side's square, leading to an incorrect cosine value and thus an incorrect angle.
3. Forgetting the inverse trigonometric function: After calculating \( \sin A \) or \( \cos A \), students sometimes write this value as the angle itself, rather than applying \( \sin^{-1} \) or \( \cos^{-1} \).
   • Wrong answer produced: An angle value that is typically between -1 and 1 (for cosine/sine values) instead of a value in degrees. For example, stating \( A = 0.5 \) when \( \sin A = 0.5 \) means \( A = 30^\circ \).

30-second recap

The Sine Rule is for when you have a side-angle pair and one other piece of information. The Cosine Rule is for when you have three sides (SSS) or two sides and the included angle (SAS). Always label your triangle carefully and ensure you're using the correct formula for finding a side or an angle.